High Quality Content by WIKIPEDIA articles! Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory that states: Every partially ordered set, in which every chain (i.e. totally ordered subset) has an upper bound, contains at least one maximal element.It is named after the mathematicians Max Zorn and Kazimierz Kuratowski.The terms are defined as follows. Suppose (P,?) is a partially ordered set. A subset T is totally ordered if for any s, t in T we have either s ? t or t ? s. Such a set T has an upper bound u in P if t ? u for all t in T. Note that u is an element of P but need not be an element of T. A maximal element of P is an element m in P such that for no element x in P, m < x.Zorn's lemma is equivalent to the well-ordering theorem and the axiom of choice, in the sense that any one of them, together with the Zermelo–Fraenkel axioms of set theory, is sufficient to prove the others.

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